The British government could only ensure the protection of European citizens living in the UK if British citizens were afforded the same rights in other EU states, Theresa May has said in the presence of the Italian prime minister.

The British prime minister’s remarks came during a press conference with Matteo Renzi, who said that the UK’s vote to leave the European Union represented a blow to 600,000 Italians living there.

Asked by a journalist whether the rights of Italians living in the UK could be protected, May said: “I want to be able to guarantee their rights in the UK. I expect to be able to do that and I intend to be able to do that, to guarantee their rights. The only circumstances in which that would not be possible would be if the rights of British citizens living in other EU member states were not guaranteed.”

It was not clear whether the issue was discussed in depth between the two leaders, or whether May’s remarks represented a concern for British citizens abroad, or a veiled threat to counterparts like Renzi who want to see ensure their citizens are not adversely affected.

May and Renzi’s meeting appeared to be cordial. Standing under the blazing sun in Villa Doria Pamphili with St Peter’s dome in the distance, both promised to continue to cooperate on issues ranging from terrorism to trade, with Renzi singling out Italian defence firm Leonardo (formerly Finmeccanica) as a point of shared interest.

But the meeting did not apparently convince Renzi to tone down his call for the UK to offer more guidance on the timing of its formal departure from the EU. Renzi said a “clear timeline” from the UK was necessary for those countries who remain in the trading bloc.

Even as May emphasised the UK’s £24bn trading relationship with Italy and its “close economic ties”, the prime minister reiterated her claim that the UK launch of Article 50, which will formally begin the exit procedure, required more time, so that all parties could work out the “nature of our relationship”.

Though Renzi faces an increasingly sceptical Italian public, with the populist and anti-euro Five Star Movement gaining in popularity, the former mayor of Florence remains one of the most vocal proponents of the EU in Europe and, according to analyst Federico Santi of the Eurasia Group, is keen to settle Brexit-related uncertainty in financial markets that could harm Italy.

“I think Italy has walked a middle ground between France and Germany in the sense that they haven’t been as harsh as [French president François] Hollande, but has been somewhat less accomodating than [German chancellor Angela] Merkel,” Santi said.

While it is still too early to tell whether Italy will emerge as a tough negotiating adversary once the UK does formally move ahead with Brexit, Renzi indicated that he did not think Britain should be offered much wiggle room when it comes to the issue of access to the single market and freedom of movement.

May said the government “will deliver” on British voters’ demand for more controls on freedom of movement, but that she would approach the issue with an open mind, given the necessity to also deliver the best possible trading deal with the EU in the future.

Renzi suggested all politics required compromise, but at the same time emphasised that the Brexit vote would have consequences and that a failure to deliver those consequences would represent a problem of credibility.